Tensor product of coherent sheaves on projective n-space

Question

I am thinking about the functor $-\otimes \mathcal F$, where $\mathcal F$ is a coherent $\mathcal O_{\mathbb P^n}$ module. The setting I am working in is the Grothendieck group of $\mathbb P^n$ and I want to define a product on it via some Tor terms. In any case, I need the functor above to go from coherent $\mathcal O_{\mathbb P^n}$ modules to coherent $\mathcal O_{\mathbb P^n}$ modules. Do I know this in general?

I know this in the affine case but I am sure it does not work for an arbitrary scheme $X$. But do I know this for $\mathbb P^n$?

Sincerely, slin0

Answer 1

The stacks project is really vast. For some reason I couldn't find this yesterday but for anyone looking for a reference for my question, [Tag 0AZR] in the stacks project deals with all of this.